## The Kurosh-Ore exchange property.(English)Zbl 0675.06003

For a lattice L we denote by J(L) the set of all join irreducibles of L; next let $$J_ 0(L)=J(L)\cup \{O_ L\}$$. If $$a\in L$$, $$a=x_ 1\vee x_ 2\vee...\vee x_ n$$ and $$x_ i\in J_ 0(L)$$ for $$i=1,2,...,n$$, then $$x_ 1\vee...\vee x_ n$$ is said to be a $$\vee$$-representation of a. The Kurosh-Ore exchange property for $$\vee$$-representations will be denoted by KOP. J. P. S. Kung [Order 2, 105-112 (1985; Zbl 0582.06008)] investigated the notion of consistent lattices. The present author proves the following theorem: In a lattice L, in which each element has a $$\vee$$-representation, the KOP holds if and only if L is consistent. Next, the author studies the following form of KOP for a closure structure (X,Cl): If A,B$$\subseteq X$$ and $$Cl(A)=Cl(B)$$, then for each $$a\in A$$ there exists $$b\in B$$ such that $$Cl(A)=Cl(A\setminus \{a\})\cup \{b\}$$. In the last section of the paper, several conditions for a finite lattice L are found which are equivalent with the condition saying that L is semimodular and has the KOP.
Reviewer: J.Jakubík

### MSC:

 06B05 Structure theory of lattices 06C10 Semimodular lattices, geometric lattices

Zbl 0582.06008
Full Text:

### References:

 [1] P. Crawley, Decomposition theory for nonsemimodular lattices,Trans. Amer. Math. Soc.,99 (1961), 246–254. · Zbl 0098.02601 [2] P. Crawley, R. P. Dilworth,Algebraic theory of lattices, Prentice-Hall (1973). · Zbl 0494.06001 [3] P. H. Edelman, Meet-distributive lattices and the antiexchange closure,Alg. Universalis,10 (1980), 290–299. · Zbl 0442.06004 [4] P. H. Edelman, R. E. Jamison,The theory of convex geometries. Preliminary version of a survey article (1984). [5] U., Faigle, Geometries on partially ordered sets,Journal of combinatorial theory,28 (1980), 26–51. · Zbl 0416.05029 [6] G. Grätzer,General lattice theory, Birkhäuser Verlag (1978). · Zbl 0436.06001 [7] C. Hermann,S-verklebte Summen von Verbänden,Math. Z.,130 (1973), 255–274. · Zbl 0275.06007 [8] J. P. S. Kung, Matchings and Radon transforms in lattices. I. Consistent lattices,Order 2 (1985), 105–112. · Zbl 0582.06008 [9] J. P. S. Kung, Radon transforms in combinatorics and lattice theory. To appear in: I. Rival (ed.),Combinatorics and Ordered Sets, Contemporary Math., Amer. Math. Soc., Providence, Rhode Island (1985). [10] J. P. S. Kung, Private communication. [11] H. Mehrtens, Die Entstehung der Verbandstheorie. arbor scientiarum. Beiträge zur Wissenschaftsgeschichte, Reihe A Abhandlungen, Gerstenberg Verlag Hildesheim (1979). · Zbl 0434.06001 [12] K. Reuter, Matching for linearly indecomposable modular lattices,Discr. Math. 63 (1987), 245–247. · Zbl 0608.06007 [13] R. Wille, Complete tolerance relations of concept analysis. Contributions to general algebra 3. Proc. of the Vienna Conf. June 21–24, 1984, 397–415. [14] R. Wille, Subdirect decomposition of concept lattices,Alg. Universalis,17 (1983), 275–287. · Zbl 0539.06006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.